Positive form

In complex geometry, the term positive form refers to several classes of real differential forms of Hodge type (p, p).

(1,1)-forms

Real (p,p)-forms on a complex manifold M are forms which are of type (p,p) and real, that is, lie in the intersection :${\displaystyle \Lambda ^{p,p}(M)\cap \Lambda ^{2p}(M,{\mathbb {R} }).}$ A real (1,1)-form ${\displaystyle \omega }$ is called positive if any of the following equivalent conditions hold

1. ${\displaystyle -\omega }$ is an imaginary part of a positive (not necessarily positive definite) Hermitian form.
2. For some basis ${\displaystyle dz_{1},...dz_{n}}$ in the space ${\displaystyle \Lambda ^{1,0}M}$ of (1,0)-forms, ${\displaystyle {\sqrt {-1}}\omega }$ can be written diagonally, as ${\displaystyle {\sqrt {-1}}\omega =\sum _{i}\alpha _{i}dz_{i}\wedge d{\bar {z}}_{i},}$ with ${\displaystyle \alpha _{i}}$ real and non-negative.
3. For any (1,0)-tangent vector ${\displaystyle v\in T^{1,0}M}$, ${\displaystyle -{\sqrt {-1}}\omega (v,{\bar {v}})\geq 0}$
4. For any real tangent vector ${\displaystyle v\in TM}$, ${\displaystyle \omega (v,I(v))\geq 0}$, where ${\displaystyle I:\;TM\mapsto TM}$ is the complex structure operator.

Positive line bundles

In algebraic geometry, positive (1,1)-forms arise as curvature forms of ample line bundles (also known as positive line bundles). Let L be a holomorphic Hermitian line bundle on a complex manifold,

${\displaystyle {\bar {\partial }}:\;L\mapsto L\otimes \Lambda ^{0,1}(M)}$

its complex structure operator. Then L is equipped with a unique connection preserving the Hermitian structure and satisfying

${\displaystyle \nabla ^{0,1}={\bar {\partial }}}$.

This connection is called the Chern connection.

The curvature ${\displaystyle \Theta }$ of a Chern connection is always a purely imaginary (1,1)-form. A line bundle L is called positive if

${\displaystyle {\sqrt {-1}}\Theta }$

is a positive definite (1,1)-form. The Kodaira embedding theorem claims that a positive line bundle is ample, and conversely, any ample line bundle admits a Hermitian metric with ${\displaystyle {\sqrt {-1}}\Theta }$ positive.

Positivity for (p, p)-forms

Positive (1,1)-forms on M form a convex cone. When M is a compact complex surface, ${\displaystyle dim_{\mathbb {C} }M=2}$, this cone is self-dual, with respect to the Poincaré pairing :${\displaystyle \eta ,\zeta \mapsto \int _{M}\eta \wedge \zeta }$

For (p, p)-forms, where ${\displaystyle 2\leq p\leq dim_{\mathbb {C} }M-2}$, there are two different notions of positivity. A form is called strongly positive if it is a linear combination of products of positive forms, with positive real coefficients. A real (p, p)-form ${\displaystyle \eta }$ on an n-dimensional complex manifold M is called weakly positive if for all strongly positive (n-p, n-p)-forms ζ with compact support, we have ${\displaystyle \int _{M}\eta \wedge \zeta \geq 0}$.

Weakly positive and strongly positive forms form convex cones. On compact manifolds these cones are dual with respect to the Poincaré pairing.

References

• P. Griffiths and J. Harris (1978), Principles of Algebraic Geometry, Wiley. ISBN 0-471-32792-1
• J.-P. Demailly, L² vanishing theorems for positive line bundles and adjunction theory, Lecture Notes of a CIME course on "Transcendental Methods of Algebraic Geometry" (Cetraro, Italy, July 1994).